ENTANGLEMENT IN QUANTUM LIQUIDS & GASES

Transcription

1 ENTANGLEMENT IN QUANTUM LIQUIDS & GASES Understanding how quantum information is encoded in quantum matter H. Barghathi UVM E. Casiano-Diaz UVM C. Herdman UVM IQC Middlebury P.-N. Roy U Waterloo R.G. Melko U Waterloo and PI Adrian Del Maestro University of Vermont

2 Quantum Entanglement emergent spacetime quantum matter quantum technologies

3 J 5 N/2. In an interferometric sequence, the of rotations in which one of the rotation angle measured. A sufficient criterion for the inp quantum-enhanced metrology, is given j2s ~2J DJz2 =(hjx i2 zhjy i2 ) is the squeezing pa ref. 6. The fluctuations of the spin in one direc below shot noise (here DJz2 vj =2), and the s orthogonal plane, ÆJxæ2 1 ÆJyæ2, has to be large sensitivity of the interferometer. A pictorial condition is shown in Fig. 1b.pThe ffiffiffiffi precisio = N, whereas enhanced measurement is jps ffiffiffiffi limit set by shot noise is 1= N. In this Letter, we report the observation states in a Bose Einstein condensate of 87R are distributed over a small number of latt and six) in a one-dimensional optical lattic pation number per site ranges from 100 to modes supporting the squeezing are two state motion corresponding to the condensate me in the standard quantum limit, which is reached in today s best available sensors of various quantities such as time2 and position3,4. Many of these sensors are interferometers in which the standard quantum limit can be overcome by using quantumentangled states (in particular spin squeezed states5,6) at the two input ports. Bose Einstein condensates of ultracold atoms are considered good candidates to provide such states involving a large number of particles. Here we demonstrate spin squeezed states suitable for atomic interferometry by splitting a condensate into a few parts using a lattice potential. Site-resolved detection of the atoms allows the measurement of the atom number difference and relative phase, which are conjugate variables. The observed fluctuations imply entanglement between the particles7 9, a resource that would allow a precision gain of 3.8 db over the standard quantum limit for interferometric measurements. Spin squeezing was one of the first quantum strategies proposed to overcome the standard quantum limit, in a precision measurement5,6 that triggered many experiments It applies to measurements where the final readout is done by counting the occupancy difference Quantum liquids and gases week ending 18 APRIL 2014 HYSICAL REVIEW LETTERS Multiparticle Entanglement of Dicke States Theory: focused on systems with discrete Hilbert spaces with local degrees of freedom: qubits, insulating lattice models, e Vitagliano,2 Jan Arlt,3 Luis Santos,4 Géza Tóth,2,5,6 and Carsten Klempt1 niz Universität Hannover, Welfengarten 1, D Hannover, Germany niversity of the Basque Country UPV/EHU, P.O. Box 644, E Bilbao, Spain Fysik og Astronomi, Aarhus Universitet, 8000 Århus C, Denmark Leibniz Universität Hannover, Appelstraße 2, D Hannover, Germany, Basque Foundation for Science, E Bilbao, Spain cs, Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary ved 27 February 2014; published 17 April 2014) 5.%&'60&-7.$2('$1(%()% 0 "#$$G./H+,$-)&.'/0&-)* Experiments employ the quantum and itinerant positional 0()%&'60&-7.$2('$2.'3-()% trate the production of many thousands of neutral atoms entangled in their present a criterion for estimating the amount of entanglement based on a. It outperforms previous criteria and applies to a wider class of entangled Experimentally, we produce a Dicke-like state using spin dynamics in a criterion proves that it contains at least genuine 28-particle entanglement. ng parameter of 11:4ð5Þ db. states of ultra-cold atomic gasses and BECs guing features of gredient for many ence [1,2], quananced metrology mber of particles tomography [6], of photons [7,8], circuits [10,11]. atrix is hindered al increase of the urthermore, it is dividual particles rticles occupy the glement of manyy measuring the e components of i si, the sum of all a quantified by means of the so-called entanglement depth, defined as the number of particles in the largest nonseparable subset [see Fig. 1(a)]. There have been numerous experiments detecting multiparticle entanglement involving up to 14 qubits in systems, where the particles can be addressed individually [9,20 24]. Large ensembles of neutral atoms observation and manipulation of Dicke states parameter ξ2 ¼ class of spinlity can be used t, since all spinlarge clouds of prepared in such al gas cells [13], in Bose-Einstein (c) separable (a) c &-3.$& b R. Islam, et.al., Nature (2015) 5.7 µm Jˆz N cos f Jˆx 10 Jx Jz Jy Jeff <J ^ / J max 0.8 Jˆy 1 > ultra high-precision B. Lücke, et.al., PRL 112, (2014) B)FB$%&.E% B)$%&.E% quantum interferometry eff FIG. 1 (color online). Measurement of the entanglement depth for a total number of 8000 atoms. (a) The entanglement depth is given by the number of atoms in the largest nonseparable subset (shaded areas). (b) The spins of the individual atoms add up to the total spin J whose possible orientations can be depicted on the Bloch sphere. Dicke states are represented by a ring around the equator with an ultralow width ΔJ z and a large radius J eff. (c) The entanglement depth in the vicinity of a Dicke state can be inferred from a measurement of these values. The red lines indicate the boundaries for various entanglement depths. The experimental result is shown as blue uncertainty ellipses with 1 and 2 standard deviations, proving an entanglement depth larger than 28 (dashed line). Figure 1 Observing spin squeezing in a Bose Einstein condensate confined in a double- or six-well trap. a, The atoms are trapped in an optical lattice potential superimposed on an harmonic dipole trap. The number of occupied sites is adjusted by changing the confinement in the lattice direction. High-resolution imaging allows us to resolve each site. b, Gain in quantum.estève, metrology is etobtained al., for spin squeezed states exhibiting reduced fluctuations in one direction (z) and a sufficiently large polarization in the Nature 1216on(2008) orthogonal plane (x,455, y) as depicted the Bloch sphere. For our system, spin fluctuations in the z direction translate to atom number difference fluctuations Dn between two adjacent wells. The polarization of the spin in $$""#$6$<$@>A$$ $$$$$$$$$$$$$$$ $ boson sampling exhibit more than anglement is best under the terms of nse. Further distrio the author(s) and and DOI. C. Shen, et al., PRL 112, (2014) Published by American Physical Society 1 1 π n 0.6 f Jz d entropy in lattice gases 250 (b) 23 Rényi Counts PACS numbers: d, Bg, Mn, Mn ( Jz ) n N 2 the x y plane is proportional to the phase coheren wells. c, The atom number fluctuations at each si integrating the atomic density obtained from abso compare a typical histogram showing sub-poisson atom number difference with the binomial distrib green curve corresponds to the deduced distribut photon shot noise, leading to a number squeezing d, The phase coherence is inferred from the interf adjacent wells. The histogram shown corresponds Æcos wæ multiparticle entanglement $$$$"""#$6$<$B>A$ $$$$$$ $$ of trapped ions Kirchhoff-Institut fu r Physik, Universita t Heidelberg, Im Neuenheimer Feld 227, Heidelberg, Germany T. Monz, et.al., PRL 102, (2009)

4 How does quantum indistinguishability affect entanglement?

5 Can we use the entanglement in quantum fluids as a resource for information processing?

6 How does entanglement scale with the size of the subregion?

7 How does entanglement scale with the size of the subregion?

8 How does entanglement scale with the size of the subregion?

9 How does entanglement scale with the size of the subregion?

10 How does entanglement scale with the size of the subregion? x y

11 How does entanglement scale with the size of the subregion?

12 How does entanglement scale with the size of the subregion?

13 How does entanglement scale with the size of the subregion?

14 How does entanglement scale with the size of the subregion? A Ā S() ~ " " =?

15 Entanglement and Entropy quantifying uncertainty in many-body systems A Measuring Entanglement SWAP algorithm in experiment and quantum Monte Carlo Results for Quantum Liquids & Gases benchmarking, scaling and the area law

16 Toy Quantum Matter bosons with hard-cores on a 1d lattice L = 4 sites N = 2 bosons H = kinetic Äb b + b ä b potential + V n n +1 Investigate the quantum ground state for different interaction strengths V

17 What are the ground states? V " 1 solid T = U = V Ä b ä b +1 + h.c. n n +1 i = 1 p 2 ( 1010i i) V 1 # superfluid i = 1 2 ( 1010i i)+ 1 2 p 2 ( 1100i i i i)

18 A Quantum Bipartition Break up the system into two parts and make a local measurement on Ā A Ā Suppose we find: what do we know about A? V " 1 no uncertainty = complete knowledge V 1 uncertainty = incomplete knowledge

19 Entanglement quantum information that is encoded non-locally in the joint state of a system Can it be quantified? Can it be measured?

20 Quantifying Entanglement with Entropy Entropy: A measure of encoded information Entanglement: Non-locally encoded quantum information Entanglement Entropy: A measure of entanglement S = probabilities encoded in reduced density matrix p log p S(A) = Tr A log A Shannon von Neumann A Ā A Ā S(A) = 0 S(A) > 0 S(A) : how entangled are A and Ā

21 Rényi Entanglement Entropies Rényi An alternate measure of entanglement S (A) = 1 og Tr 1 A lim 1 S (A) = Tr A log A 0.8 S1(A) A Ā Rényi Entropy S $ (A) S1(A) # S2(A) # S3(A) # Rényi Index ($)

22 How does quantum indistinguishability affect entanglement?

23 Bipartitions of identical particles Different ways to partition ground state Mode Bipartition Constructed from the Fock space of single-particle modes Ā A i = n A nā c na nā n A nā A S(A) Particle Bipartition Label a subset of n particles na = n i = r nb = N - n Z 1,...,r N i n = n-body density matrix dr n+1 dr N h i n S(n)

24 Example: a simple quantum liquid I 1d Bose-Hubbard model H = J Ä b j b j+1 + b j+1 b j ä + U 2 n j n j 1 j j U/J 0 superfluid 3.3 insulator example: L = 2, N = 2 spatial bipartition: A Ā A = TrĀ = i = 20i + 11i + 02i 2 n=0 hn ih ni Ā Ā = A S 1 ( A )= Tr A ln A = 2 ln 2 2 ln 2 2 ln 2

25 Example: a simple quantum liquid II 1d Bose-Hubbard model H = J Ä b j b j+1 + b j+1 b j ä + U 2 n j n j 1 j j U/J 0 superfluid 3.3 insulator example: L = 2, N = 2 particle bipartition: A Ā i = i + p 2 ( i i) i 1 = 2 =1 h2 ih 2 i = p p A

26 Example: a simple quantum liquid III 1d Bose-Hubbard model H = J Ä b j b j+1 + b j+1 b j ä + U 2 n j n j 1 j j U/J 0 superfluid 3.3 insulator example: L = 2, N = 2 different bipartitions can provide complimentary information on phases, interactions & statistics entanglement entropy U/J spatial particle

27 Can we use the entanglement in quantum fluids as a resource for information processing?

28 Or is it all just fluffy bunnies? J. Dunningham, A. Rau, and K. Burnett, Science 307, 872 (2005) Using entanglement as a resource requires ability to perform local physical operations on subsystems Particle Entanglement inaccessible due to the indistinguishability of particles N. Killoran, M. Cramer, and M. B. Plenio, PRL 112, (2014) N = 2, L = 4 Spatial Entanglement particle number conservation prohibits swapping all entanglement to register H. M. Wiseman and J. A. Vaccaro, PRL 91, (2003) x1 =?, x2 =? ïå ã ò SWAP 1i A 1i B + 2i A 0i B + 0i A 2i B 0i reg

29 Operational entanglement Get around these difficulties by combining the two measures. H. M. Wiseman and J. A. Vaccaro, PRL 91, (2003) S op (A) = n P n S(A n ) N - n A n = 1 P n ˆP n probability A ˆP n projection operator n Maximal amount of entanglement that can S op (A) <S(A) S op (A) > 0 ) S(n) > 0 be produced between quantum registers by local operations.

30 Operational Entanglement Example example: L = 2 i = 20i + 11i + 02i = ih = 2 2 S op 1 (A) = A 2 = 2 n A A 1 = 1 A A = TrĀ = P n S 1 (A n ) A n = 1 P n ˆP n A ˆP n A A 0 = A Ā A A S op 1 (A) =0 need at least 2 states in the subsystem

31 Experimental Measurement Density matrix is generally inaccessible = 0i = 1i C. F. Roos et al, PRL 92, (2004) Re ρ Re % 0.5 (a) 00i i = p 1 ( 01i + 10i) 2 Im ρ 01i 0.5 Im % 10i 11i > 00> > 10> > 10> 11> 11> 00> 01> 10> 11> 00> 01> 10> 11> Re ρ (b) Measurement becomes exponentially difficult Im ρ 4 particles on 4 sites: % ~ 10 5 entries

32 Entanglement and Entropy quantifying uncertainty in many-body systems A Measuring Entanglement SWAP algorithm in experiment and quantum Monte Carlo Results for Quantum Liquids & Gases benchmarking, scaling and the area law

33 The Replica Method Computing Rényi entanglement entropies by swapping subregions between non-interacting identical copies α = 2 2 replicas of system swap subregions Ā A Tr A A 0 = = = =, 0, 0, 0, 0 \SWAP claim: S 2 (A) = log Tr 2 A = log b,b 0 b,b 0 b,b 0 A Ā 0 0 A 0 A b 0 b 0 b 0 0 b 0 b 0 b b b 0 A = b2ā b 0 b 0 0 SWAP b 0 b 0 = Tr î Ā hb bi Ā A A 0 SWAP ó D \ SWAPE expectation value in replicated ensemble SWAP 0 P. Calabrese and J. Cardy, J. Stat. Mech.: Theor. Exp. P06002 (2004)

34 0 0 Rényi entropies can be computed via local expectation values Tr ρa 2 0 0

35 Experimental Measurement C. Hong, Z. Ou, and L. Mandel, PRL , (1987) A. J. Daley, H. Pichler, J. Schachenmayer, and P. Zoller, Phys. Rev. Lett. 109, (2012) R. Islam, R. Ma, P. M. Preiss, M. E. Tai, A. Lukin, M. Rispoli, and M. Greiner, Nature 528, 77 (2015) IDEA: Use bosonic Hong-Ou-Mandel interference with atoms 30)) 45"$/6 H = J j Ä b ä j b j+1 + b j+1 b j + U 2 n j n j 1 j insulator ' ( superfluid ( * ' * U/Jx

36 Exact Diagonalization Results H = J j Ä b ä j b j+1 + b j+1 b j + U 2 n j n j 1 j \ L = 12 experiment L = 6 U/J superfluid U/J Mott R. Melko, C. Herdman, D. Iouchtchenko, P.-N. Roy and A.D. Phys. Rev. A, 93, (2016)

37 Exact Diagonalization Results H = J j Ä b ä j b j+1 + b j+1 b j + U 2 n j n j 1 j L = 12 L = 6 S op 2 = n P n S 2 (A n ) Operational entanglement peaks near the quantum phase transition superfluid U/J Mott R. Melko, C. Herdman, D. Iouchtchenko, P.-N. Roy and A.D. Phys. Rev. A, 93, (2016)

38 Exact diagonalization is limited to small systems with discrete Hilbert spaces Can we get bigger?

39 Path Integral Ground State QMC Description H = N =1 2 2m T (R2M ) 2 ˆ + N =1 V + <j U j N interacting particles in d-dimensions & 0 (RM ) Configurations projecting a trial wavefunction to the H i Ti e = lim ground state 0 gives discrete imaginary time worldlines constructed from products of the short 0 H 0 R time propagator G(R, R ; ) = R e Observables exact method for computing ground state expectation values O = h T e h H O e T e 2 H H Ti Ti T (R0 ) L Updates Local and non-local bead updates with weights given by '()

40 Porting the Replica Method to PIGS Break paths at the center time slice &, measure when replicas are linked via short time propagator G. Ā space T(R) 0 (R) T(R) SWAP A $ Ā N = 6 SWAP A $ Ā SWAP = Ä G R R 0, SWAP î R + R 0 + ó ; ä Technology adapted from other QMC flavors M. B. Hastings, I. González, A. B. Kallin, and R. G. Melko, PRL 104, (2010) R. Melko, A. Kallin, and M. Hastings, PRB 82, (2010) C. Herdman, R. Melko and A.D. Phys. Rev. B, 89, (2014) C. M. Herdman, S. Inglis, P. N. Roy, R. G. Melko, and A.D., PRE 90, (2014) T. Grover, Phys. Rev. Lett. 111, (2013) Assaad, Lang, Toldin, Phys. Rev. B 89, (2014) Broecker and Trebst, J. Stat. Mech. (2014) P08015 J. E. Drut and W. J. Porter, PRB 92, (2015)

41 Entanglement and Entropy quantifying uncertainty in many-body systems A Measuring Entanglement SWAP algorithm in experiment and quantum Monte Carlo Results for Quantum Liquids & Gases benchmarking, scaling and the area law

42 Benchmarking on a Solvable Model Lieb-Liniger model of (-function interacting bosons on a ring H = 1 2 N =1 d 2 d 2 + g <j ( j) A N = 2 QMC ratio trick weak interactions N = 8 gl N strong interactions E. H. Lieb and W. Liniger, PR 130, 1605 (1963) C. M. Herdman, P. N. Roy, R. G. Melko, and A.D., PRB B 94, (2016)

43 Scaling of Spatial Entanglement Expected scaling result for 1d critical systems: apple L sin 1 S (,L)= c log L + c + O p A } D(N,l) c % 1 P. Calabrese and J. Cardy, J. Stat. Mech. P06002 (2004) J. Cardy and P. Calabrese, J. Stat. Mech. P (2010) C. M. Herdman, P. N. Roy, R. G. Melko, and A.D., PRB B 94, (2016)

44 Particle Entanglement Predicted general scaling form by Haque & Schoutens: N S(n, N) = ln + b n O. Zozulya, M. Haque, and K. Schoutens, PRA 78, (2008) M. Haque, O. S. Zozulya, and K. Schoutens, J. Phys. A 42, (2009) na = 3 For a bosonic Luttinger Liquid: a = n/k S 2 (n = 1) S 2 (n = 2) C. Herdman and A.D., Phys. Rev. B, 91, (2015) confirmation with QMC Luttinger parameter Bethe Ansatz

45 Sensitivity of Entanglement to Statistics spatial bipartition: 1d critical systems apple L sin 1 S (,L)= c log L + c + O p P. Calabrese and J. Cardy, J. Stat. Mech. P06002 (2004) J. Cardy and P. Calabrese, J. Stat. Mech. P (2010) particle bipartition: S (n, N) = (n) ln N n + b (n)+o N 1 (n) a = 1 for fermions = n/k for 1d bosons C. Herdman and A.D., Phys. Rev. B, 91, (2015) H. Barghathi, E. Casiano-Diaz, and AD, JSTAT. 2017, (2017) M. Haque, O. S. Zozulya, and K. Schoutens, J. Phys. A 42, (2009)

46 entanglement area law? How does entanglement scale with the size of the subregion in d>1? S() ~ " thermodynamic entropy is A Ā extensive " = d S 8S 2

47 Black Hole Entropy Area Law Black hole thermodynamics: Quantum black holes emit thermal radiation Area Law: entropy of a black hole is proportional to surface area, not volume SBH " area J.D. Bekenstein, PRD 7, 2333 (1973) S.W. Hawking, Nature 248, 30 (1974) Is this due to entanglement? Toy model: coupled harmonic oscillators Area Law: number of springs connecting A with Ā scales with A R boundary size M. Srednicki Phys. Rev. Lett. 71, 666 (1993)

48 General Derivation of the Area Law Based on 2 physical principles: 1. S(A) arises from correlations local to the entangling surface at scale r: local S(A, r) = Z A (d-1) surface element ds g( A, r) rd 1 quantity depending on curvature of *A *A A 2. All length scales contribute: microscopic to macroscopic S(A) = = Z R Z R r 0 d(log r)s(a, r) r 0 = c 0 2 dr r R r 0 Z A ds r 2 apple 2 + c 1 log R c 0 + c 1 Å r R ã 2 + r 0 + const. + O area law for a sphere spherical *A g = c 0 + c 1 Å r R 1 R 2 ã 2 + c 2 Å r no odd powers M.B. Hastings, J. Stat. Mech., P08024 (2007) S.N. Solodukhin, Phys. Lett. B, 693, 605 (2010) B. Swingle, ariv: H. Liu and M. Mezai, JHEP 1304, 162 (2013) L. Hayward Sierens, Ph.D. Thesis, (2017) R ã 4

49 Temperature What about a real quantum phase of matter? helium-4 solid Pressure superfluid normal fluid gas

50 Entanglement in Superfluid 4 He 3d box at T = 0 with periodic boundary conditions at SVP V(r) Ĥ = N =1 2 2m ˆ 2 + N =1 ˆV + <j Û j r Measure entanglement S2(R) between spherical region of radius R and the rest of the box R Investigate scaling by changing the radius of the sphere C. M. Herdman, P.-N. Roy, R. G. Melko & A.D. Nature Phys 13, 556 (2017)

51 Scaling of the Entanglement S2(R) 4R 2 C. M. Herdman, P.-N. Roy, R. G. Melko & A.D. Nature Phys 13, 556 (2017)

52 Area: Area Law vs. Volume Law R 2 Volume: 2 = 4 + c S ƒ t r 0 S ƒ t 2 = 4 3 R r c 2 fit parameters inter-particle separation C. M. Herdman, P.-N. Roy, R. G. Melko & A.D. Nature Phys 13, 556 (2017)

53 Entanglement in quantum liquids can be useful Physical constraints determine the entanglement that can be transferred to a register for quantum information processing Discovery of an area law in a real quantum liquid Quantum entanglement scales with the surface area and not volume in superfluid 4 He S2 R 2 Analogous to Bekenstein-Hawking black hole entropy

54 Partners in Research & Computing